Simpsons Quadrature Formula

For the numerical evaluation of expectation values, numerical quadrature schemes are needed. In the case of atoms (see chapter 9), these are onedimensional and, hence, particularly simple. As an example, we discuss Simpson s rule in the following. 

The famous Simpson rule for numerical quadrature can be written as [479, p.l02] 

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Some quadrature formula, such as Simpson s rule, gives the solution faster than does numerical integration of the differential equation by a general method. 

A better solution is to use one of the many existing quadrature formulas—algebraic expressions that provide estimates of the integrals of tabulated data. Several such formulas are presented and illustrated in Appendix A.3 the use of one of them, Simpson s rule, is required for the integration of heat capacity data in several problems at the end of this chapter. 

The many existing quadrature formulas differ only in the choice of functions to fit to the data points. Two of the simplest approximations are to fit straight lines between successive points and sum the area under the lines, and to fit parabolas to successive triplets of points and sum the areas under the parabolas. These approximations lead to the quadrature formulas known respectively as the trapezoidal rule and Simpson s rule. We will discuss each in turn. 

When you have an analytical expression for an integrand y(x) but you cannot perform the integration from jr = a to -c = b analytically, the procedure is to evaluate y at a series of values of X from a to b—that is, to generate a data table—and then to use a quadrature formula such as Simpson s rule to estimate the integral. Now, however, you have the choice of the number of evaluations of y(x) to make. 

Figure A-4 Simpson s four-point rule illustratioa 4. Five-point quadrature formula ...

Some recent Runge-Kutta formulae are based on quadrature methods, that is, the points at which the intermediate stage approximations are taken are the same points used in integration with either Gauss or Lobatto or Radau rules (Chapter 1). For example, the Runge-Kutta method derived from the Lobatto quadrature with three points (also called the Cavalieri-Simpson rule) is... 

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